ADVENT: a published Goldilocks design

Trial overview

ADVENT was a multicenter, prospective, single-blind, randomized controlled noninferiority trial. Randomized subjects were assigned 1:1 to PFA or standard-of-care thermal ablation, where the thermal arm used either radiofrequency ablation or cryoballoon ablation depending on site. The first 1 to 3 subjects at each site were nonrandomized roll-in subjects and are not part of the randomized comparison modeled below.

The final published randomized cohort included 305 subjects assigned to PFA and 302 assigned to thermal ablation. In the primary results paper, PFA was noninferior to thermal ablation for both primary effectiveness and primary safety.

The published design also included a trial-flow figure. The following diagram recreates the parts of that flow that matter for the package mapping.

In the code below, roll-in subjects and randomized subjects excluded from the analysis population are ignored. The simulation starts directly with the modeled 1:1 analysis population.

The Goldilocks flow

The ADVENT design paper includes a flowchart for the adaptive sample size algorithm. The chart below recreates the logic in package terms rather than copying the published image.

The corresponding goldilocks arguments are:

N_total <- 750
interim_look <- c(350, 450, 550, 650)

Sn <- c(0.95, 0.90, 0.85, 0.80)
Fn <- c(0.05, 0.10, 0.10, 0.10)

The mapping is direct:

• N_total = 750 is the maximum modeled mITT analysis-population size.
• interim_look = c(350, 450, 550, 650) supplies the mITT accrual milestones at which the Goldilocks decision rule is evaluated. The maximum sample size is not included in interim_look.
• Sn contains the published predicted-success thresholds for sample sizes 350, 450, 550, and 650.
• Fn contains the published futility thresholds for the same looks.

ADVENT required the predictive probability rule to be favorable for both co-primary endpoints before enrollment stopped for predicted success, and unfavorable for either endpoint before stopping for futility. The current survival_adapt() interface models one endpoint per run. In this vignette we therefore run the effectiveness and safety endpoints separately, then explain how those separate endpoint-specific runs relate to the reported co-primary design.

The SAP allowed up to 900 enrolled subjects to obtain 750 mITT subjects: up to 750 mITT subjects, 105 nonrandomized roll-in subjects, and 45 randomized subjects who did not enter the mITT population. The package examples below start directly with the analysis population and do not simulate roll-ins or post-randomization exclusions.

Endpoint scale

ADVENT reported the primary effectiveness endpoint as treatment success at 12 months. The beta-binomial method in goldilocks is parameterized on the binary event probability. For the effectiveness endpoint we therefore code the event as failure by 12 months:

\[ p_{\text{failure}} = 1 - p_{\text{treatment success}}. \]

The design paper reports a target scenario with 65% treatment success in each arm. On the event scale used in the code, that corresponds to a 35% failure probability in each arm.

For the safety endpoint, the event is already an adverse event. The target scenario was an 8% primary safety event rate in each arm.

The ADVENT primary analyses used Bayesian beta-binomial endpoint models with noninformative \(\operatorname{Beta}(0.5, 0.5)\) priors. In goldilocks, this is specified with:

bin_prior <- c(0.5, 0.5)

The prior argument still appears in the code below because survival_adapt() uses a time-to-event model to impute not-yet-observed outcomes at interim looks. That imputation model uses a Gamma prior on piecewise-exponential hazards. The ADVENT SAP specifies a noninformative \(\operatorname{Gamma}(0.5, 0.001)\) shape-rate prior for most of those hazards. We use those reported hyperparameters here rather than the package default. Because the SAP expresses the corresponding risk sets and exposure in patient-days, every time supplied to goldilocks below is expressed in days:

prior <- c(0.5, 0.001)

That distinction is important: bin_prior is the ADVENT-aligned endpoint prior, whereas prior controls the package’s predictive imputation model.

The effectiveness model in the SAP needs one further qualification. It partitions follow-up at days 90, 104, 150, and 210. For interim sample-size prediction, the first four interval hazards use \(\operatorname{Gamma}(0.5, 0.001)\) priors, but the 210–360-day hazard uses an informative \(\operatorname{Gamma}(5, 10000)\) prior because little late follow-up was expected at the interim assessments. The final multiple-imputation analysis returns to \(\operatorname{Gamma}(0.5, 0.001)\) for every interval. The current goldilocks interface applies one shared Gamma prior to every interval and uses it for both interim prediction and final imputation, so the special 210–360-day prior is not yet available in the package. We therefore use \(\operatorname{Gamma}(0.5, 0.001)\) throughout this endpoint-specific illustration.

Later follow-up intervals can have no exposure at an early look. The SAP’s conjugate model then leaves the corresponding hazard governed by its prior. The package’s historical default instead propagates sufficient statistics from a neighboring interval, so the examples explicitly set empty_interval = "prior".

The SAP reports \(M = 5000\) completed datasets for its predictive-probability and multiple-imputation calculations. The evaluated parts of this vignette deliberately use fewer imputations: 50 for the worked single trials and 30 for the operating-characteristic example. We do not use \(M = 5000\) in evaluated code in order to keep vignette computation time to a minimum. The unevaluated fuller template does use \(M = 5000\) to show the ADVENT setting. Consequently, the numerical results shown in the vignette are demonstrations of package workflow rather than reproductions of the ADVENT calibration.

SAP event-time models and time units

The SAP provides the exact default piecewise-exponential generators used in its operating-characteristic simulations. Because their hazards are expressed per patient-day and their Gamma prior rates use patient-day exposure, this vignette uses days as the single numeric time unit supplied to goldilocks. The SAP also treats 30 days as one simulation month, so 12 months is represented as 360 days.

days_per_month <- 30L
follow_up_months <- 12L
end_of_study_day <- follow_up_months * days_per_month

eff_event_cutpoints_day <- c(90, 104, 150, 210)
eff_hazard_per_day <- c(
  0.000111670,
  0.002197976,
  0.003163208,
  0.002839089,
  0.000494053
)

safety_event_cutpoints_day <- 7
safety_hazard_per_day <- c(0.011137363, 1.53540e-5)

event_free_at_interval_end <- function(hazard, interval_end) {
  vapply(seq_along(hazard), function(j) {
    cutpoints <- if (j == 1L) NULL else interval_end[seq_len(j - 1L)]
    post <- array(hazard[seq_len(j)], dim = c(1L, j, 1L))

    1 - goldilocks:::haz_to_prop(
      post = post,
      cutpoints = cutpoints,
      end_of_study = interval_end[j],
      single_arm = TRUE
    )$p_treatment
  }, numeric(1))
}

implied_event_free_proportion <- c(
  event_free_at_interval_end(
    eff_hazard_per_day,
    c(eff_event_cutpoints_day, end_of_study_day)
  ),
  event_free_at_interval_end(
    safety_hazard_per_day,
    c(safety_event_cutpoints_day, end_of_study_day)
  )
)

hazard_table <- data.frame(
  Endpoint = c(rep("Effectiveness failure", 5), rep("Safety event", 2)),
  `Follow-up interval (days)` = c(
    "0--<90", "90--<104", "104--<150", "150--<210",
    "210--360", "0--<7", "7--360"
  ),
  `Hazard per patient-day` = c(
    eff_hazard_per_day,
    safety_hazard_per_day
  ),
  `Implied event-free proportion at interval end` =
    implied_event_free_proportion,
  check.names = FALSE
)

knitr::kable(hazard_table, digits = c(9, 3))

The less-than sign marks the exclusive upper boundary of each nonfinal interval; the final interval includes the administrative endpoint at day 360. This notation makes clear that adjacent windows do not overlap. Because event time is continuous, assigning an exact cut-point to the interval on its left or right would not change the model probability.

The following calculation verifies the two final binary event probabilities used by the vignette:

prob_check <- data.frame(
  Endpoint = c("Effectiveness failure", "Safety event"),
  `SAP target event probability at day 360` = c(0.35, 0.08),
  `Calculated event probability at day 360` = c(
    ppwe(
      hazard = matrix(eff_hazard_per_day, nrow = 1),
      end_of_study = end_of_study_day,
      cutpoints = eff_event_cutpoints_day
    ),
    ppwe(
      hazard = matrix(safety_hazard_per_day, nrow = 1),
      end_of_study = end_of_study_day,
      cutpoints = safety_event_cutpoints_day
    )
  ),
  check.names = FALSE
)

knitr::kable(prob_check, digits = 6)

For a piecewise-exponential model with interval hazards \(\lambda_j\) and interval lengths \(d_j\), the cumulative hazard through day 360 is

\[ H(360) = \sum_j \lambda_j d_j, \]

and the implied day-360 event probability is

\[ p = 1 - \exp\{-H(360)\}. \]

To obtain a target event probability \(p^*\) while retaining the relative height of every piece, multiply each hazard by

\[ c = \frac{-\log(1-p^*)}{H(360)}, \qquad \lambda_j^* = c\lambda_j. \]

When the treatment hazard vector is obtained by scaling the control vector this way, the hazard ratio is the same constant \(c\) in every interval. Thus, the operation preserves the piecewise shape and imposes proportional hazards in data generation. We use it for the noninferiority-boundary examples below:

scale_pwe_to_event_probability <- function(
  hazard_per_day,
  cutpoints_day,
  end_day,
  target_event_probability
) {
  interval_length_day <- diff(c(0, cutpoints_day, end_day))
  cumulative_hazard <- sum(hazard_per_day * interval_length_day)
  scale <- -log1p(-target_event_probability) / cumulative_hazard
  hazard_per_day * scale
}

eff_margin_hazard_per_day <- scale_pwe_to_event_probability(
  eff_hazard_per_day,
  eff_event_cutpoints_day,
  end_of_study_day,
  target_event_probability = 0.50
)

safety_margin_hazard_per_day <- scale_pwe_to_event_probability(
  safety_hazard_per_day,
  safety_event_cutpoints_day,
  end_of_study_day,
  target_event_probability = 0.16
)

Two time clocks remain conceptually distinct even though both are represented in days. Event cut-points are subject-relative follow-up times; enrollment-rate change points below are trial-calendar times measured from first patient in. In addition, the SAP’s event clock starts on the index-procedure date, whereas goldilocks starts it at enrollment/randomization. This vignette treats those dates as coincident and does not model a randomization-to-procedure delay.

Effectiveness endpoint

For the effectiveness endpoint, success in the code means:

\[ \Pr(p_{\text{PFA failure}} - p_{\text{thermal failure}} < 0.15 \mid \text{data}) > 0.956. \]

This matches the reported 15 percentage point absolute noninferiority margin for treatment success, after translating success into failure. A lower failure probability is better, so we use alternative = "less".

ADVENT used 1:1 randomization stratified by site. The design paper reports randomly varying permuted block sizes, but the SAP does not provide the block sizes or the randomization-generation algorithm. The package can generate blocked 1:1 randomization, but it does not model site-level stratification or randomly varying block sizes. Here block = 2 and rand_ratio = c(1, 1) retain simple 1:1 balance; they do not reproduce the trial’s full randomization scheme.

The SAP explicitly designates 5% of subjects as lost to follow-up for safety and 7.5% for effectiveness: the same 5% plus another 2.5% who may be unassessable for effectiveness while remaining assessable for safety. SAP dropout times are uniform over days 0–360, and an event remains observed when it precedes dropout. The package’s prop_loss mechanism is similar but not identical: every selected subject is censored at a uniformly generated time before their potential event or administrative censoring time. Separate endpoint runs also cannot preserve the SAP’s shared subject-level 5% plus nested 2.5% relationship. We therefore use 0.075 and 0.05 to map the SAP’s designated fractions, not to claim exact reproduction of its censoring process. The subsequently observed proportion not completing follow-up was approximately 0.04, which is close to 0.05, but that post-trial quantity is not the design input and need not equal endpoint-specific missingness.

The SAP reports the expected accrual schedule in subjects per month: 2, 5, 10, 15, 20, 25, and 30 during months 1–7, then 33 per month from month 8 onward. Accrual is into the primary analysis population. The source rates remain visibly labeled per month below, but they are divided by 30 exactly once before being supplied to goldilocks. Likewise, the calendar-month change points are converted once to days since first patient in. The SAP does not describe its individual arrival-time generator; goldilocks uses these rates in a continuous-time Poisson enrollment process with first patient in fixed at calendar day 0.

enrollment_rate_per_month <- c(2, 5, 10, 15, 20, 25, 30, 33)
enrollment_rate_change_month <- 1:7

enrollment_rate_per_day <- enrollment_rate_per_month / days_per_month
enrollment_rate_change_day <-
  enrollment_rate_change_month * days_per_month

effectiveness_prop_loss <- 0.075
safety_prop_loss <- 0.05

enrollment_schedule <- data.frame(
  `Trial-calendar interval (months)` = c(
    paste("Month", 1:7),
    "Month 8 onward"
  ),
  `Trial-calendar interval (days since first patient in)` = c(
    paste0(
      (0:6) * days_per_month,
      "--<",
      (1:7) * days_per_month
    ),
    paste0(7 * days_per_month, " onward")
  ),
  `SAP rate (subjects/month)` = enrollment_rate_per_month,
  `Rate supplied to goldilocks (subjects/day)` = enrollment_rate_per_day,
  check.names = FALSE
)

knitr::kable(enrollment_schedule, digits = 4)

Thus, for example, trial-calendar day 210 is the beginning of month 8 and the 33-subject/month rate. It is unrelated to the effectiveness endpoint’s subject-relative cut-point at follow-up day 210. The 360-day follow-up horizon is also per subject, not the total duration of the trial. Accrual speed remains worth sensitivity checking: faster accrual reduces the amount of observed endpoint information available at interim looks.

The following survival_adapt() call simulates and analyzes the effectiveness endpoint for one simulated trial. It does not estimate operating characteristics over multiple trials.

set.seed(4601)

# One simulated effectiveness-endpoint trial
advent_effectiveness <- survival_adapt(
  hazard_treatment = eff_hazard_per_day,
  hazard_control = eff_hazard_per_day,
  cutpoints = eff_event_cutpoints_day,
  N_total = N_total,
  lambda = enrollment_rate_per_day,
  lambda_time = enrollment_rate_change_day,
  interim_look = interim_look,
  end_of_study = end_of_study_day,
  prior = prior,
  bin_prior = bin_prior,
  bin_method = "quadrature",
  block = 2,
  rand_ratio = c(1, 1),
  prop_loss = effectiveness_prop_loss,
  alternative = "less",
  h0 = 0.15,
  Fn = Fn,
  Sn = Sn,
  prob_ha = 0.956,
  N_impute = 50,
  empty_interval = "prior",
  method = "bayes-bin",
  imputed_final = TRUE
)

advent_effectiveness
#>   prob_threshold margin alternative N_treatment N_control N_enrolled N_max
#> 1          0.956   0.15        less         225       225        450   750
#>   post_prob_ha     est_final ppp_success stop_futility stop_expected_success
#> 1    0.9994649 -0.0007079646        0.98             0                     1

The most important columns are:

• N_enrolled: the selected sample size for this simulated trial.
• stop_expected_success: whether enrollment stopped because the current sample size appeared adequate.
• stop_futility: whether enrollment stopped because success looked unlikely even at the maximum sample size.
• est_final: the posterior mean of \(p_{\text{PFA failure}} - p_{\text{thermal failure}}\).
• post_prob_ha: the final posterior probability that the binary endpoint effect is below the noninferiority margin.

Safety endpoint

The safety endpoint uses the same Goldilocks sample-size rule, but changes the event probability, margin, and posterior probability threshold. Success in the code means:

\[ \Pr(p_{\text{PFA safety event}} - p_{\text{thermal safety event}} < 0.08 \mid \text{data}) > 0.966. \]

This survival_adapt() call likewise represents one simulated trial for the safety endpoint.

set.seed(4602)

# One simulated safety-endpoint trial
advent_safety <- survival_adapt(
  hazard_treatment = safety_hazard_per_day,
  hazard_control = safety_hazard_per_day,
  cutpoints = safety_event_cutpoints_day,
  N_total = N_total,
  lambda = enrollment_rate_per_day,
  lambda_time = enrollment_rate_change_day,
  interim_look = interim_look,
  end_of_study = end_of_study_day,
  prior = prior,
  bin_prior = bin_prior,
  bin_method = "quadrature",
  block = 2,
  rand_ratio = c(1, 1),
  prop_loss = safety_prop_loss,
  alternative = "less",
  h0 = 0.08,
  Fn = Fn,
  Sn = Sn,
  prob_ha = 0.966,
  N_impute = 50,
  empty_interval = "prior",
  method = "bayes-bin",
  imputed_final = TRUE
)

advent_safety
#>   prob_threshold margin alternative N_treatment N_control N_enrolled N_max
#> 1          0.966   0.08        less         325       325        650   750
#>   post_prob_ha  est_final ppp_success stop_futility stop_expected_success
#> 1    0.9988901 0.01815951           1             0                     1

In the published ADVENT design, a predicted-success stopping recommendation required high predictive probability for both endpoints. A futility recommendation could be triggered by low predictive probability for either endpoint. The two separate simulations above do not impose that joint rule. Instead, they show how each endpoint-specific Bayesian rule maps to survival_adapt().

A compact design object

For simulations, it is helpful to collect the settings that genuinely are common and then add endpoint-specific cut-points and designated missingness fractions. Keeping those endpoint-specific arguments out of the common object prevents an effectiveness setting from being reused accidentally for safety, or vice versa.

advent_common <- list(
  N_total = N_total,
  lambda = enrollment_rate_per_day,
  lambda_time = enrollment_rate_change_day,
  interim_look = interim_look,
  end_of_study = end_of_study_day,
  prior = prior,
  bin_prior = bin_prior,
  bin_method = "quadrature",
  block = 2,
  rand_ratio = c(1, 1),
  alternative = "less",
  Fn = Fn,
  Sn = Sn,
  N_impute = 30,
  empty_interval = "prior",
  method = "bayes-bin",
  imputed_final = TRUE,
  ncores = 2
)

advent_effectiveness_args <- modifyList(advent_common, list(
  cutpoints = eff_event_cutpoints_day,
  prop_loss = effectiveness_prop_loss
))

advent_safety_args <- modifyList(advent_common, list(
  cutpoints = safety_event_cutpoints_day,
  prop_loss = safety_prop_loss
))

advent_effectiveness_args
#> $N_total
#> [1] 750
#> 
#> $lambda
#> [1] 0.06666667 0.16666667 0.33333333 0.50000000 0.66666667 0.83333333 1.00000000
#> [8] 1.10000000
#> 
#> $lambda_time
#> [1]  30  60  90 120 150 180 210
#> 
#> $interim_look
#> [1] 350 450 550 650
#> 
#> $end_of_study
#> [1] 360
#> 
#> $prior
#> [1] 0.500 0.001
#> 
#> $bin_prior
#> [1] 0.5 0.5
#> 
#> $bin_method
#> [1] "quadrature"
#> 
#> $block
#> [1] 2
#> 
#> $rand_ratio
#> [1] 1 1
#> 
#> $alternative
#> [1] "less"
#> 
#> $Fn
#> [1] 0.05 0.10 0.10 0.10
#> 
#> $Sn
#> [1] 0.95 0.90 0.85 0.80
#> 
#> $N_impute
#> [1] 30
#> 
#> $empty_interval
#> [1] "prior"
#> 
#> $method
#> [1] "bayes-bin"
#> 
#> $imputed_final
#> [1] TRUE
#> 
#> $ncores
#> [1] 2
#> 
#> $cutpoints
#> [1]  90 104 150 210
#> 
#> $prop_loss
#> [1] 0.075

Then the endpoint-specific simulation calls become short:

set.seed(4610)

eff_target <- do.call(sim_trials, c(
  advent_effectiveness_args,
  list(
    N_trials = 500,
    hazard_treatment = eff_hazard_per_day,
    hazard_control = eff_hazard_per_day,
    h0 = 0.15,
    prob_ha = 0.956,
    return_trace = TRUE,
    seed = 4610
  )
))

eff_null_boundary <- do.call(sim_trials, c(
  advent_effectiveness_args,
  list(
    N_trials = 500,
    hazard_treatment = eff_margin_hazard_per_day,
    hazard_control = eff_hazard_per_day,
    h0 = 0.15,
    prob_ha = 0.956,
    seed = 4611
  )
))

oc_small <- summarise_sims(list(
  "target: equal 35% failure" = eff_target$sims,
  "margin: PFA failure 50%" = eff_null_boundary$sims
))

knitr::kable(oc_small, digits = 3)

Each scenario uses 500 simulated trials and two cores. This remains a workflow demonstration rather than a definitive estimate of power or type I error. For design work, increase N_trials, increase N_impute, and run sensitivity analyses over accrual speed, loss to follow-up, event rates, and the imputation hazard model.

Visualizing the simulated design

The plotting helpers answer complementary design questions. First, plot_sim_ocs() compares operating characteristics across the target and noninferiority-margin scenarios. The true PFA event probability is supplied as the effect scale.

oc_small$true_pfa_event_probability <- c(0.35, 0.50)
plot_sim_ocs(
  oc_small,
  effect = "true_pfa_event_probability",
  xlab = "True 12-month PFA event probability"
)

Within the target scenario, plot_sim_stopping() shows how often each sample size is selected and why enrollment stops.

plot_sim_stopping(eff_target)

Because eff_target was simulated with return_trace = TRUE, its interim decision geometry can also be inspected. Each displayed panel corresponds to an ADVENT enrollment milestone reached in at least one simulated trial; larger bubbles represent predictive-probability pairs reached by more simulated trials.

plot_sim_decisions(eff_target)

A fuller simulation template

The following code is closer to what one would run outside the vignette build. It is not evaluated here.

advent_effectiveness_full_args <- modifyList(advent_effectiveness_args, list(
  N_impute = 5000,
  ncores = 8
))

advent_safety_full_args <- modifyList(advent_safety_args, list(
  N_impute = 5000,
  ncores = 8
))

eff_target_full <- do.call(sim_trials, c(
  advent_effectiveness_full_args,
  list(
    N_trials = 1000,
    hazard_treatment = eff_hazard_per_day,
    hazard_control = eff_hazard_per_day,
    h0 = 0.15,
    prob_ha = 0.956,
    seed = 4620
  )
))

eff_margin_full <- do.call(sim_trials, c(
  advent_effectiveness_full_args,
  list(
    N_trials = 1000,
    hazard_treatment = eff_margin_hazard_per_day,
    hazard_control = eff_hazard_per_day,
    h0 = 0.15,
    prob_ha = 0.956,
    seed = 4621
  )
))

safety_target_full <- do.call(sim_trials, c(
  advent_safety_full_args,
  list(
    N_trials = 1000,
    hazard_treatment = safety_hazard_per_day,
    hazard_control = safety_hazard_per_day,
    h0 = 0.08,
    prob_ha = 0.966,
    seed = 4622
  )
))

safety_margin_full <- do.call(sim_trials, c(
  advent_safety_full_args,
  list(
    N_trials = 1000,
    hazard_treatment = safety_margin_hazard_per_day,
    hazard_control = safety_hazard_per_day,
    h0 = 0.08,
    prob_ha = 0.966,
    seed = 4623
  )
))

summarise_sims(list(
  "effectiveness target" = eff_target_full$sims,
  "effectiveness margin" = eff_margin_full$sims,
  "safety target" = safety_target_full$sims,
  "safety margin" = safety_margin_full$sims
))

The “margin” scenarios above set the treatment event probability equal to the control event probability plus the noninferiority margin. They are useful for checking false-positive behavior near the design boundary. They are not the only null scenarios worth studying.

SAP sensitivity assumptions and reference benchmarks

The SAP assessed whether operating characteristics were robust to several alternative assumptions. In goldilocks notation, these correspond to:

• Varying the day-360 event probability implied by the effectiveness hazard_control vector over c(0.40, 0.35, 0.30). These are failure probabilities and therefore correspond to the SAP’s control success probabilities of 0.60, 0.65, and 0.70. The day-360 event probability implied by the safety hazard_control vector was varied over c(0.06, 0.08, 0.10).
• Supplying a slower lambda schedule of c(2, 5, 10, 10, 15, 15, 20, 25) / 30, with lambda_time = (1:7) * 30, or a faster schedule of c(5, 10, 20, 25, 30, 40, 50) / 30, with lambda_time = (1:6) * 30. Division by 30 converts the SAP’s subjects/month rates to the subjects/day scale used here.
• Using the SAP’s second effectiveness generator with cutpoints = c(90, 100, 200, 300) and both hazard_treatment and hazard_control set to c(0.000055695, 0.010034797, 0.001690763, 0.000680535, 0.000573122) in its equal-arm target scenario.
• Using the SAP’s third effectiveness generator with cutpoints = c(90, 120) and both hazard arguments set to c(0.000569925, 0.007879626, 0.000596254) in its equal-arm target scenario.
• Using the SAP’s second safety generator with cutpoints = c(7, 180) and both hazard arguments set to c(0.007327613, 0.000122991, 0.0000600610) in its equal-arm target scenario.

The three alternative event-time generators deliberately use data-generating partitions that differ from the principal analysis partitions. The current package accepts one cutpoints vector for both event generation and predictive imputation, so it cannot reproduce those model-misspecification sensitivities directly. The SAP also planned an imputation-model sensitivity with six pieces: day 90 followed by five post-90-day intervals containing approximately equal event counts, potentially with different cut-points by arm. A shared fixed package cut-point vector cannot reproduce that analysis either.

The SAP supplies quantitative reference results based on 10,000 simulated trials per scenario and \(M=5000\) predictive draws or imputations:

The target row sets the effectiveness hazard_treatment and hazard_control vectors to imply equal day-360 failure probabilities of 0.35, and the safety hazard vectors to imply equal day-360 event probabilities of 0.08; success requires both co-primary endpoints. At the effectiveness boundary, hazard_treatment implies a 0.50 failure probability while hazard_control implies 0.35, and the safety endpoint was assumed always to recommend stopping for promise and to pass finally. At the safety boundary, the treatment and control safety hazards imply event probabilities of 0.16 and 0.08, respectively, and effectiveness was treated analogously. These are external validation targets. The endpoint-separated simulations in this vignette do not claim to reproduce them.

References

Broglio KR, Connor JT, Berry SM. Not too big, not too small: a Goldilocks approach to sample size selection. Journal of Biopharmaceutical Statistics. 2014;24(3):685-705. doi:10.1080/10543406.2014.888569.

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